2017/05/09 by Gendron, Quentin, Tahar, Guillaume · 1 citation
#Algebraic Geometry (math.AG) #FOS: Mathematics #Geometric Topology (math.GT)
paper · doi:10.48550/arxiv.1705.03240
We study the local invariants that a meromorphic k-differential on a Riemann surface of genus g≥0 can have. These local invariants are the orders of zeros and poles, and the k-residues at the poles. We show that for a given pattern of orders of zeroes, there exists, up to a few exceptions, a primitive k-differential having these orders of zero. The same is true for meromorphic k-differentials and in this case, we describe the tuples of complex numbers that can appear as k-residues at their poles. For genus g≥2, it turns out that every expected tuple appears as k-residues. On the other hand, some expected tuples are not the k-residues of a k-differential in some remaining strata. This happens in the quadratic case in genus 1 and in genus zero for every k. We also give consequences of these results in algebraic and flat geometry.